$Letf\left(x\right)=\sin x+\cos x,g\left(x\right)=x^2-1.Theng\left(f\left(x\right)\right)isinvertibleforx\in$

$\begin{array}{l}f\left(x\right)=\sin x+\cos x,g\left(x\right)=x^2-1\\ g\left(f\left(x\right)\right)=g(\sin x+\cos x)\\ ={(\sin x+\cos x)}^2-1\\ ={\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x-1\\ =1+2\sin x\cos x-1\\ =\sin 2x\\ Weknowthat-1\le \sin 2x\le 1\forall 2x\in [\frac{-\pi }{2},\frac{\pi }{2}]\\ \Rightarrow x\in [\frac{-\pi }{4},\frac{\pi }{4}]\end{array}$